Robust data-driven control for nonlinear systems using the Koopman operator

Abstract

Data-driven analysis and control of dynamical systems have gained a lot of interest in recent years. While the class of linear systems is well studied, theoretical results for nonlinear systems are still rare. In this paper, we present a data-driven controller design method for discrete-time control-affine nonlinear systems. Our approach relies on the Koopman operator, which is a linear but infinite-dimensional operator lifting the nonlinear system to a higher-dimensional space. Particularly, we derive a linear fractional representation of a lifted bilinear system representation based on measured data. Further, we restrict the lifting to finite dimensions, but account for the truncation error using a finite-gain argument. We derive a linear matrix inequality based design procedure to guarantee robust local stability for the resulting bilinear system for all error terms satisfying the finite-gain bound and, thus, also for the underlying nonlinear system. Finally, we apply the developed design method to the nonlinear Van der Pol oscillator.

arXiv commentAccepted for presentation at the IFAC World Congress 2023

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• An overview of Koopman-based control: From error bounds to closed-loop guarantees
• Control of bilinear systems using gain-scheduling: Stability and performance guarantees
• SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems
• Decrypting Nonlinearity: Koopman Interpretation and Analysis of Cryptosystems
• Koopman-based control of nonlinear systems with closed-loop guarantees
• Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency
• Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators
• Robust Control Design and Analysis Based on Lifting Linearization of Nonlinear Systems Under Uncertain Initial Conditions